## Advances in Differential Equations

- Adv. Differential Equations
- Volume 3, Number 4 (1998), 575-593.

### Existence and uniqueness of positive solutions to certain differential systems

Patricio Felmer and Salomé Martínez

#### Abstract

In this article we study existence and uniqueness of positive solutions for elliptic systems of the form $$ \begin{align} -\Delta v = &f(x,u) \quad \hbox{in} \quad \Omega \\ -\Delta u = & v^\beta \quad \hbox{in}\quad \Omega, \end{align} $$ with Dirichlet boundary condition on a bounded smooth domain in $\Bbb R^N$. The nonlinearity $f$ is assumed to have a sub-$\beta$ growth with $\beta>0$, that in case $f(x,u)=u^\alpha, \alpha>0$, corresponds to $\alpha\beta<1$. The results are also valid for a larger class of systems, including some infinite dimensional Hamiltonian Systems.

#### Article information

**Source**

Adv. Differential Equations, Volume 3, Number 4 (1998), 575-593.

**Dates**

First available in Project Euclid: 18 April 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1366292564

**Mathematical Reviews number (MathSciNet)**

MR1659242

**Zentralblatt MATH identifier**

0946.35028

**Subjects**

Primary: 35J55

Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35J65: Nonlinear boundary value problems for linear elliptic equations

#### Citation

Felmer, Patricio; Martínez, Salomé. Existence and uniqueness of positive solutions to certain differential systems. Adv. Differential Equations 3 (1998), no. 4, 575--593. https://projecteuclid.org/euclid.ade/1366292564